🔷 Penrose tilings

What you’re seeing

The floor is covered by two shapes: a thick diamond (gold) with angles of 72° and 108°, and a thin one (green) with angles of 36° and 144°. All edges are the same length.

Press Cut once more and every tile is cut into smaller thick and thin tiles by the same fixed rule. The new tiles are smaller by the golden ratio $\varphi \approx 1.618$. Do it again and again. The pattern spreads to fill the view, with stars, flowers and rings of ten, but it never settles into a repeating grid like bathroom tiles do.

Watch the stats line: the number of thick tiles divided by the number of thin tiles gets closer and closer to $\varphi$.

Why it works

Half-tiles make the rule simple. Each diamond is cut along a diagonal into two triangles, called Robinson triangles. A thin diamond gives two triangles with angles 36°, 72°, 72°. A thick diamond gives two with angles 108°, 36°, 36°. The cutting rule (called deflation) is:

• a thin half-tile becomes one smaller thin half and one smaller thick half;
• a thick half-tile becomes one smaller thin half and two smaller thick halves.

The new points are placed $1/\varphi$ of the way along an edge. Since $\varphi$ is the ratio of a diagonal to a side in a regular pentagon, all the angles still fit.

Counting the tiles. Write $t_n$ and $T_n$ for the thin and thick half-tiles at level $n$. The rule says

$$t_{n+1} = t_n + T_n, \qquad T_{n+1} = t_n + 2T_n.$$

These are Fibonacci-style steps. The ratio $T_n / t_n$ heads to the number $r$ with $r = \frac{1 + 2r}{1 + r}$, which gives $r^2 = r + 1$, so $r = \varphi = \frac{1 + \sqrt{5}}{2}$.

Why it can’t repeat. If the tiling repeated like wallpaper, a large enough patch would hold thick and thin tiles in a fixed ratio of whole numbers. But in this tiling the ratio is exactly $\varphi$, and $\varphi$ is irrational: it is not a fraction of whole numbers. So no repeating pattern can match it. (The same two diamonds can tile a floor in a repeating way if you place them freely. Penrose added matching rules, marks on the edges that must line up, and with those rules every tiling of the whole plane is non-repeating.)

History. Roger Penrose found his tilings in the 1970s. The best known use a kite and a dart, or these two diamonds. In 1982 Dan Shechtman found crystals whose atoms are ordered like this, with fivefold symmetry that a repeating crystal cannot have. These quasicrystals won him the 2011 Nobel Prize in Chemistry. In 2023, David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss found a single tile shape, nicknamed “the hat,” that covers the plane only in non-repeating ways.

Try this

• At level 1, find a thin and a thick diamond. Measure (by eye) their angles. Do 36° + 144° and 72° + 108° each make 180°?
• Count the tiles yourself for levels 0 to 3 using the rule above, starting from 10 thin halves. Do your numbers match the stats line?
• At level 6, look for a “star” of five thick diamonds meeting at a point. How many can you find?
• Pause the animation midway through a fade and see the old tiles behind the new ones.